Foam lecture #1 - Boulder School for Condensed Matter and ...

THE PHYSICS OF FOAM
• Boulder School for Condensed Matter and Materials Physics
July 1-26, 2002: Physics of Soft Condensed Matter
1. Introduction
Formation
Microscopics
2. Structure
Experiment
Simulation
3. Stability
Coarsening
Drainage
4. Rheology
Linear response
Rearrangement & flow
Douglas J. DURIAN
UCLA Physics &
Astronomy
Los Angeles, CA 90095-
1547

Foam is…
• …a random packing of bubbles in a relatively small
amount of liquid containing surface-active impurities
– Four levels of structure:
– Three means of time evolution:
• Gravitational drainage
• Film rupture
• Coarsening (gas diffusion from smaller to larger bubbles)

• …a most unusual form of condensed matter
–Like a gas:
• volume ~ temperature / pressure
– Like a liquid:
• Flow without breaking
• Fill any shape vessel
– Under large force, bubbles rearrange their packing configuration
– Like a solid:
• Support small shear forces elastically
– Under small force, bubbles distort but don’t rearrange
Foam is…

• Everyday life:
– detergents
– foods (ice cream, meringue, beer, cappuccino, ...)
– cosmetics (shampoo, mousse, shaving cream, tooth paste, ...)
• Unique applications:
– firefighting
– isolating toxic materials
– physical and chemical separations
– oil recovery
– cellular solids
• Undesirable occurrences:
– mechanical agitation of multicomponent liquid
– pulp and paper industry
– paint and coating industry
– textile industry
– leather industry
– adhesives industry
– polymer industry
– food processing (sugar, yeast, potatoes)
– metal treatment
– waste water treatment
– polluted natural waters
Foam is…
• familiar!
•important!
• need to control stability and mechanics
• must first understand microscopic structure and dynamics…

Condensed-matter challenge
• To understand the stability and mechanics of bulk foams
in terms of the behavior at microscopic scales
bubbles are the “particles” from which foams are assembled
– Easy to relate surfactant-film and film-bubble behaviors
– Hard to relate bubble-macro behavior
• Opaque: no simple way to image structure
• Disordered: no periodicity
• k BT

• Similar challenge for seemingly unrelated systems
Jamming
– Tightly packed collections of bubbles, droplets, grains, cells,
colloids, fuzzy molecules, tectonic plates,.…
• jammed/solid-like: small-force / low-temperature / high-density
• fluid/liquid-like: large-force / high-temperature / low-density
force-chains (S. Franklin) avalanches (S.R. Nagel) universality?

Foam Physics Today
• visit the websites of these Summer 2002 conferences to
see examples of current research on aqueous foams
– Gordon Research Conference on Complex Fluids
• Oxford, UK
– EuroFoam 2002
• Manchester, UK
– Foams and Minimal Surfaces
• Isaac Newton Institute for Mathematical Sciences
– Geometry and Mechanics of Structured Materials
• Max Planck Institute for the Physics of Complex Systems
• after these lectures, you should be in a good position to understand
the issues being addressed & progress being made!

General references
1. D. Weaire and N. Rivier, “Soap, cells and statistics - random patterns in two dimensions,” Contemp. Phys. 25, 55 (1984).
2. J. P. Heller and M. S. Kuntamukkula, “Critical review of the foam rheology literature,” Ind. Eng. Chem. Res. 26, 318-325 (1987).
3. A. M. Kraynik, “Foam flows,” Ann. Rev. Fluid Mech. 20, 325-357 (1988).
4. J. H. Aubert, A. M. Kraynik, and P. B. Rand, “Aqueous foams,” Sci. Am. 254, 74-82 (1989).
5. A. J. Wilson, ed., Foams: Physics, Chemistry and Structure (Springer-Verlag, New York, 1989).
6. J. A. Glazier and D. Weaire, “The kinetics of cellular patterns,” J. Phys.: Condens. Matter 4, 1867-1894 (1992).
7. C. Isenberg, The Science of Soap Films and Soap Bubbles (Dover Publications, New York, 1992).
8. J. Stavans, “The evolution of cellular structures,” Rep. Prog. Phys. 56, 733-789 (1993).
9. D. J. Durian and D. A. Weitz, “Foams,” in Kirk-Othmer Encyclopedia of Chemical Technology, 4 ed., edited by J.I. Kroschwitz
(Wiley, New York, 1994), Vol. 11, pp. 783-805.
10. D. M. A. Buzza, C. Y. D. Lu, and M. E. Cates, “Linear shear rheology of incompressible foams,” J. de Phys. II 5, 37-52 (1995).
11. R. K. Prud'homme and S. A. Khan, ed., Foams: Theory, Measurement, and Application. Surfactant Science Series 57, (Marcel
Dekker, NY, 1996).
12. J.F. Sadoc and N. Rivier, Eds. Foams and Emulsions (Kluwer Academic: Dordrecht, The Netherlands, 1997).
13. D. Weaire, S. Hutzler, G. Verbist, and E. Peters, “A review of foam drainage,” Adv. Chem. Phys. 102, 315-374 (1997).
14. D. J. Durian, “Fast, nonevolutionary dynamics in foams,” Current Opinion in Colloid and Interface Science 2, 615-621 (1997).
15. L.J. Gibson and M.F. Ashby, Cellular Solids: Structure and Properties (Cambridge University Press, Cambridge, 1997).
16. M. Tabor, J. J. Chae, G. D. Burnett, and D. J. Durian, “The structure and dynamics of foams,” Nonlinear Science Today (1998).
17. D. Weaire and S. Hutzler, The Physics of Foams (Clarendon Press, Oxford, 1999).
18. S.A. Koehler, S. Hilgenfeldt, and H.A. Stone, "A generalized view of foam drainage,” Langmuir 16, 6327-6341 (2000).
19. A.J. Liu and S.R. Nagel, eds., Jamming and Rheology (Taylor and Francis, New York, 2001).
20. J. Banhart and D. Weaire, “On the road again: Metal foams find favor,” Physics Today 55, 37-42 (July 2002).
Thanks for the images:
John Banhart, Ken Brakke, Jan Cilliers, Randy Kamien, Andy Kraynik, John Sullivan, Paul Thomas, Denis Weaire, Eric Weeks,…

special thanks to collaborators
• Students
– Alex Gittings
– Anthony Gopal
– Pierre-Anthony Lemieux
– Rajesh Ojha
– Ian Ono
– Sidney Park
– Moin Vera
• Postdocs
– Ranjini Bandyopadhyay
– Narayanan Menon
– Corey O’Hern
– Arnaud Saint-Jalmes
– Shubha Tewari
– LoicVanel
• Colleagues
– Chuck Knobler
– Steve Langer
– Andrea Liu
– Sid Nagel
– Dave Pine
– Joe Rudnick
– Dave Weitz

• Shake, blend, stir, agitate, etc.
– Uncontrolled / irreproducible
– Unwanted foaming of multicomponent liquids
• Sparge = blow bubbles
– Polydisperse or monodisperse
– Uncontrolled/non-uniform liquid fraction
solution
porous plate
gas
foam
gas
Foam production I.
bubble raft
gas
foam

• in-situ release / production of gas
– nucleation
• eg CO 2 in beer
– aerosol:
• eg propane in shaving cream
• small bubbles!
–active:
• eg H 2 in molten zinc
• eg CO 2 from yeast in bread
Foam production II.

Foam production III.
• turbulent mixing of thin liquid jet with gas
– vast quantities
– small polydisperse bubbles
– controlled liquid fraction
• lab samples
• firefighting
• distributing pesticides/dyes/etc.
• covering landfills
• supressing dust
• …
solution
gas
jet
foam

• many materials can be similarly foamed
– nonaqueous liquids (oil, ferrofluids,…)
– polymers (styrofoam, polyurethane,…)
– metals
–glass
– concrete
• variants found in nature
– cork
– bone
– sponge
– honeycomb
Foam production IV.

• spittle bug:
• cuckoo spit / froghoppers:
• stickleback-fish’s nest
Foams produced by animals

• antifoaming agents
– prevent foaming or break an existing foam
Foam production V.
• mysterious combination of surfactants, oils, particles,…

Microscopic behavior
• look at progressively larger length scales…
– surfactant solutions
– soap films
– local equilibrium & topology

• bubbles quickly coalesce – no foam
Pure liquid
– van der Waals force prefers monotonic dielectric profile;
therefore, bubbles attract:
a b a
l
“effective interface potential”
is free energy cost per unit area:
V vdw(l)= -A/12πl 2 , A=Hamaker constant
l

Surfactant solution
• surface active agent – adsorbs at air/water interface
– head: hydrophilic (eg salt)
– tail: hydrophobic (eg hydrocarbon chain)
surface tension, σ
• lore for good foams…
– chain length: short enough that the surfactant is soluble
– concentration: just above the “critical micelle concentration”
eg sodium dodecylsulfate (SDS)
80 erg/cm 2
CMC
30 erg/cm 2
Log [surfactant concentation, c]
{NB: lower σ doesn’t stabilize the foam…

Electrostatic “double-layer”
• adsorbed surfactants dissociate, cause repulsion necessary to
overcome van der Waals and hence stabilize the foam
– electrostatic
– entropic (dominant!)
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free energy cost per unit area:
V DL(l)= (64k BTρ/K D)Exp[-K Dl],
ρ = electrolyte concentration
K D -1 ~ ρ -1/2 = Debye screening length
– NB: This is similar to the electrostatic stabilization of colloids

Soap film tension
• film tension / interface potential / free energy per area:
γ(l) = 2σ + V VDW (l) + V DL (l) ~ 2σ
thickness, l
• disjoining pressure: Π(l) = -dγ/dl
– vanishes at equilibrium thickness, l eq ~ K D -1 (30-3000Å)

• Plateau border
– scalloped-triangular channel where three films meet
– the edge shared by three neighboring bubbles
• Vertex
– region where four Plateau borders meet
– the point shared by four neighboring bubbles
Film junctions

Liquid distribution
• division of liquid between films-borders-vertices
– repulsion vs surface tension
– wet vs dry
Π repulsion dominates
=> maximize l
dry
=> polyhedral
σ dominates
=> minimize area
wet
=> spherical

Laplace’s law
• the pressure is greater on the inside a curved interface
due to surface tension, σ = energy / area = force / length
σ P i = P o + 2σ/r {in general, ∆P= σ(1/r 1 +1/r 2 )}
r
– forces on half-sphere:
• ΣF up = P iπr 2 –P oπr 2 –2πσr = 0
– energy change = pressure x volume change:
• dU = (∆P)4πr 2 dr, where U(r)=4πr 2 σ
P o
P i
σ

Liquid volume fraction
• liquid redistributes until liquid pressure is same everywhere
bubble radius, R
• typically: film thickness l

Plateau’s rules for dry foams
• for mechanical equilibrium:
P
– i.e. for zero net force on a Plateau border,
– zero net force on a vertex,
– and Σ∆P=0 going around a closed loop:
(1) films have constant curvature & intersect three at a time at 120 o
(2) borders intersect four at a time at cos -1 (1/3)=109.47 o
• rule #2 follows from rule #1
• both are obviously correct if the films and borders are straight:
P
ΣF=0
P

Rule #1 for straight borders
• choose r 1 and orientation of equilateral triangle
• construct r 2 from extension down to axis
• construct r 3 from inscribed equilateral triangle
– NB: centers are on a line
r 1 r 3
Plateau border in/out of page
– films meet at 120 o (triangles meet at 60 o -60 o -60 o , and are normal to PB’s)
– similar triangles give (r 1+r 2)/r 1 = r 2/r 3, i.e. 1/r 1 + 1/r 2 = 1/r 3 and so ΣP=0
r 2

• proof of Plateau’s rules is not obvious!
– established in 1976 by Jean Taylor
Curved Plateau borders

Decoration theorem for wet foams
• for d=2 dimensions, an equilibrium wet foam can be
constructed by decorating an equilibrium dry foam
– can you construct an elementary proof?
• PB’s are circular arcs that join tangentially to film
• theorem fails in d=3 due to PB curvature

• periodic foam structures
• disordered foam structures
– experiment
– simulation
Next time…